`dvc exp` commands always request my passphrase.

Bug Report

dvc exp commands always request my passphrase.

Description

Hello,
I have the following unwanted behavior. First of all my config looks like this:

[core]
    remote = nas_remote
[cache]
    dir=/fastdata/data/dvc_cache
    type = symlink
[hydra]
    enabled = True
['remote "nas_remote"']
    url = ssh://192.168.93.111:/data/slidespeicher/projects/data/annotations/dvc_setup
    verify = true
    keyfile = /home/deep/.ssh/priv_key.openssh
    user = lefos
    port = 22
    ask_password = false
    ask_passphrase = true

Especially the lines

    ask_password = false
    ask_passphrase = true

are necessary because I connect to my NAS (the remote DVC cache) by using a private key, which works fine by the way. ✔️

I was working with dvc repro and I am changing to dvc exp run and I noticed that whenever I run any dvc exp command (e.g. dvc exp show or dvc exp run), I get the following prompt: Enter a passphrase for host '192.168.93.111' port '22' user 'lefos':. This is normal to be requested when I interact directly with my remote with commands like dvc pull or dvc push, but especially for commands like dvc exp run, it is annoying because it is being requested for each dvc stage.

Environment information

Output of dvc doctor:

$ dvc doctor
DVC version: 2.56.0 (pip)
-------------------------
Platform: Python 3.8.13 on Linux-5.4.0-147-generic-x86_64-with-glibc2.17
Subprojects:
        dvc_data = 0.47.5
        dvc_objects = 0.21.2
        dvc_render = 0.4.0
        dvc_task = 0.2.1
        scmrepo = 1.0.3
Supports:
        http (aiohttp = 3.8.4, aiohttp-retry = 2.8.3),
        https (aiohttp = 3.8.4, aiohttp-retry = 2.8.3),
        s3 (s3fs = 2023.5.0, boto3 = 1.26.76),
        ssh (sshfs = 2023.4.1)
Config:
        Global: /home/deep/.config/dvc
        System: /etc/xdg/dvc
Cache types: hardlink, symlink
Cache directory: ext4 on /dev/mapper/fastdatagroup-fastdatavolume
Caches: local
Remotes: ssh, s3
Workspace directory: ext4 on /dev/mapper/fastdatagroup-fastdatavolume
Repo: dvc (subdir), git
Repo.site_cache_dir: /var/tmp/dvc/repo/85b2919c00b2eaae667bc92335b08f76

Start by reproducing dvc exp show and dvc exp run with the provided SSH remote configuration and compare them with dvc pull or dvc push. Trace where the repeated SSH passphrase prompt occurs during experiment stages; done means experiment commands no longer request the passphrase once per stage while preserving remote authentication.