I can replicated this in v11.7.0 (b797). In testing I deliberately tested the tab focus switch trigger with a single ($Name) column and separately with multiple columns. Regardless of which column is adjusted, they return to default width after tab focus switch.
However, of note for @eastgate, I cannot replicate this in the current b805 beta.
[Post-edit] On checking Release Notes, this is noted as addressed in b800 RNs, so of course working in b805.
So though I don’t know the cause, it may already be fixed, albeit only in the next public release. I don’t believe there is a workaround for v11.7.0.