@typedef {Object} with function properties: Does not get past "function" (the word), ignores the types

I define a few object types for "API objects", i.e. objects returned by a function that export an "API" from within that functions scope.

Example:

/**
 * `SystemReadStream` objects are created by factory function
 * @global
 * @typedef {Object} SimpleReadStream
 * @property {function((string|ArrayBuffer)):undefined} ondata <b>Assign your event handler</b>
 * @property {function(Error):undefined} onerror <b>Assign your event handler</b>
 * @property {function():undefined} onfinished <b>Assign your event handler</b>
 * @property {function():undefined} pause Pause the read stream
 * @property {function():undefined} resume Resume the read stream
 * @property {function():undefined} close Cancel the read stream
 */

This works in WebStorm:

But from JSDoc (using the default template from 3.5.5 with four minor custom CSS rules added) I only get "function", it leaves out all the type information:

This is a function syntax used by many, just one of numerous examples (not to mention that WebStorm would not have bothered to support this if that wasn't the case): https://stackoverflow.com/a/38586423/544779

Also, it should be supported at least through the Closure Compiler types, go to "Function Type": https://github.com/google/closure-compiler/wiki/Types-in-the-Closure-Type-System or https://github.com/google/closure-compiler/wiki/Annotating-Types#function-declarations or https://github.com/google/closure-compiler/wiki/Annotating-JavaScript-for-the-Closure-Compiler#function-type

or go to https://github.com/Microsoft/TypeScript/wiki/JSDoc-support-in-JavaScript "Define function type" (Example)

So... virtually everybody says that this is the syntax to specify a function type, and the IDEs support it too. Only JSDoc doesn't?

Reproduce the issue with the supplied @typedef example, then trace JSDoc's type parsing and generated documentation output. Done means function-property parameter and return types are preserved in the output instead of being rendered only as "function".