The honest answer first
Most sources on this site benefit from conversion. This one usually does not, and it is worth saying plainly rather than burying.
HDR helps where a scene contains light the SDR range cannot hold: a sunlit window, a specular reflection, a flame. A screen recording contains none of those. It contains an interface that was designed, rendered and colour-managed for an SDR display, so its brightest pixel is diffuse white by construction. Adding headroom above diffuse white gives that headroom nothing to carry.
Where it is worth it anyway
Two cases:
The recording contains video. A screen capture of HDR footage playing back is a recording of the tone-mapped SDR version, so that is lost either way, but a capture containing ordinary video content has the usual photographic highlights and behaves like normal footage.
Delivery consistency. If a screen recording sits in a timeline alongside HDR footage, converting it avoids a format mismatch at the edit stage. That is a workflow reason rather than a picture-quality one, and it is a legitimate one.
The problem that actually hurts screen recordings
Not dynamic range: chroma subsampling. Screen content is full of hard colour edges, and coloured text on a contrasting background is the worst case for 4:2:0. The fringing and softening around text in a screen recording comes from the colour resolution being halved, and no amount of dynamic range fixes it.
If a screen recording looks bad, subsampling and the capture resolution are almost always the cause, and HDR conversion will not address either.
Checking
ffprobe -v error -select_streams v:0 \
-show_entries stream=color_transfer,pix_fmt -of default=nw=1 recording.mov
Expect bt709 and yuv420p. Screen recorders on every major platform produce SDR.
Questions people ask
What if my screen recording contains video?
Then the video portion behaves like ordinary footage and can benefit. The catch is that a capture of HDR content playing back records the tone-mapped SDR version the compositor showed, so the original HDR range is already gone before the recorder sees it.
My screen recording looks soft. Will HDR fix it?
No. Soft screen recordings are usually chroma subsampling around hard-edged coloured text, or a capture resolution below the display's native one. Both are resolution problems and dynamic range does not touch either.