`self.to_string().eq(other.to_string())` where `other: &Self` and `Self: Display` suggests removing `.to_string()` instead of taking reference

Code

#[derive(Debug)]
struct Error;

impl std::fmt::Display for Error {
    fn fmt(&self, fmt: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(fmt, "Error")
    }
}

impl std::error::Error for Error {}

// This error instead of suggesting adding an & in front of `other.to_string()`, suggests removing `.to_string()`:

// error[E0308]: mismatched types
//    --> src/lib.rs:14:29
//     |
// 14  |         self.to_string().eq(other.to_string())
//     |                          -- ^^^^^^^^^^^^^^^^^ expected `&_`, found `String`
//     |                          |
//     |                          arguments to this method are incorrect
//     |
//     = note: expected reference `&_`
//                   found struct `String`
// note: method defined here
//    --> /playground/.rustup/toolchains/stable-x86_64-unknown-linux-gnu/lib/rustlib/src/rust/library/core/src/cmp.rs:255:8
//     |
// 255 |     fn eq(&self, other: &Rhs) -> bool;
//     |        ^^
// help: try removing the method call
//     |
// 14  -         self.to_string().eq(other.to_string())
// 14  +         self.to_string().eq(other)
//     |
//

impl PartialEq for Error {
    fn eq(&self, other: &Self) -> bool {
        self.to_string().eq(other.to_string())
    }
}

// Changing to the error suggestion though:

// error[E0277]: can't compare `String` with `Error`
//   --> src/lib.rs:42:29
//    |
// 42 |         self.to_string().eq(other)
//    |                          -- ^^^^^ no implementation for `String == Error`
//    |                          |
//    |                          required by a bound introduced by this call
//    |
//    = help: the trait `PartialEq<Error>` is not implemented for `String`
//    = help: the following other types implement trait `PartialEq<Rhs>`:
//              <String as PartialEq<&'a str>>
//              <String as PartialEq<Cow<'a, str>>>
//              <String as PartialEq<str>>
//              <String as PartialEq>
//
//impl PartialEq for Error {
//    fn eq(&self, other: &Self) -> bool {
//        self.to_string().eq(other)
//    }
//}

Current output

error[E0308]: mismatched types
    --> src/lib.rs:14:29
     |
 14  |         self.to_string().eq(other.to_string())
     |                          -- ^^^^^^^^^^^^^^^^^ expected `&_`, found `String`
     |                          |
     |                          arguments to this method are incorrect
     |
     = note: expected reference `&_`
                   found struct `String`
 note: method defined here
    --> /playground/.rustup/toolchains/stable-x86_64-unknown-linux-gnu/lib/rustlib/src/rust/library/core/src/cmp.rs:255:8
     |
 255 |     fn eq(&self, other: &Rhs) -> bool;
     |        ^^
 help: try removing the method call
     |
 14  -         self.to_string().eq(other.to_string())
 14  +         self.to_string().eq(other)
     |

Desired output

error[E0308]: mismatched types
    --> src/lib.rs:14:29
     |
 14  |         self.to_string().eq(other.to_string())
     |                          -- ^^^^^^^^^^^^^^^^^ expected `&_`, found `String`
     |                          |
     |                          arguments to this method are incorrect
     |
     = note: expected reference `&_`
                   found struct `String`
 note: method defined here
    --> /playground/.rustup/toolchains/stable-x86_64-unknown-linux-gnu/lib/rustlib/src/rust/library/core/src/cmp.rs:255:8
     |
 255 |     fn eq(&self, other: &Rhs) -> bool;
     |        ^^
 help: try taking a reference
     |
 14  -         self.to_string().eq(other.to_string())
 14  +         self.to_string().eq(&other.to_string())
     |

Rationale and extra context

No response

Other cases

No response

Rust Version

From play.rust-lang.org:

Build using the Nightly version: 1.82.0-nightly (2024-08-05 e57f3090aec33cdbf660)

Anything else?

https://play.rust-lang.org/?version=nightly&mode=debug&edition=2021&gist=4fd462b1a41f891f83bfc1b66d0beef6

Reproduce the example from the issue in the Rust playground using the linked nightly version, then trace the compiler diagnostic logic that emits the “try removing the method call” suggestion for this mismatch. Done means the diagnostic suggests taking a reference to other.to_string() for the reported case without regressing the existing suggestion.