Dash – “Perhaps this is the reason why some don’t wear a tie for Mincha.”
That may be simply because of the cholent stains.
However the halachic basis is a valid one:
There is an interesting Rashi and Ran in Nedarim daf 77 that notes a custom for women to remove their Shabbat jewelry on Shabbat afternoons towards the end of the day.
(Others say that it only applies to jewelry and not to general shabbos clothes – see Orchot Chaim, OC 300:2).
The Erech Shai OC 262 says the reason one is not required to wear Shabbat clothes at this time is to recall that Yosef, Moshe, and David passed away Shabbat afternoon.
Based on this, it makes sense to be yotzei all opinions, to wear shabbos clothes to mincha but not wear the tie (which may be like jewelry).