The inch-thick dust on all the furniture, as well as the piles of unwashed clothes and dirty dishes, convinced Tony that Horace will not make a good roommate.

Has

Explanation:

The question provided is incomplete, however after checking Google I saw the full question. The question aims to check subject-verb agreement.

to have: The inch-thick dust on all the furniture, as well as the piles of unwashed clothes  and dirty dishes, convinced Tony that Horace will not make a good  roommate.

the correct form of the verb in this senternce is "has".

When a singular noun or pronoun is connected with another noun or pronoun by using words like with, as well as, besides, together with, no less than and other similar expressions, the verb is singular.

Therefore, "The inch-thick dust on all the furniture, as well as the piles of unwashed clothes and dirty dishes, has convinced Tony that Horace will not make a good roommate"