It is not correct.

In the cited post, '먹다' means "to eat food or drink", but '마음' is not food/drink. '잡수시다' and '드시다' can't be honorific expressions of "먹다" if the object is not food/drink. Instead, '먹다' can be used as "to have thoughts or emotions", so '먹으시다' is the right way to represent honor.

그분은 이제야 위기를 극복하려는 마음을 먹으셨다.

See also:

1. 먹다 in Naver Dictionary
2. Tweet from 국립국어원

It is not correct.

In the cited post, '먹다' means "to eat food or drink", but '마음' is not food/drink. '잡수시다' and '드시다' can't be honorific expressions of "먹다" if the object is not food/drink. Instead, '먹다' can be used as "to have thoughts or emotions", so '먹으시다' is the right way to represent honor.

그분은 이제야 위기를 극복하려는 마음을 먹으셨다.

cf. How about this sentence?

마음껏 드세요.

Here "마음껏" is an adverb, and "드세요" is used as an honorific expression of "먹다(to eat something)", not "먹다(to have thoughts of emotions)", so there is no grammatical error.

See also:

1. 먹다 in Naver Dictionary
2. Tweet from 국립국어원