I apologize that this is only a partial answer, but perhaps it can still be of use, since there are not yet any other answers to this question.

Since $I$ is a monomial ideal, your algebra $A$ is graded, by putting the generators $x_1, \dots ,x_n$ in degree $1$. Your algebra $A$ is furthermore *connected*, that is, it vanishes in negative degrees and in degree zero it is simply a copy of the ground field $k$ generated as a $k$-vector space by the unit element $1$. A 1983 article by George Wilson, https://www.sciencedirect.com/science/article/pii/0021869383901035 , evidently uses an elementary calculation of the Cartan map from algebraic $K_0$ to $G_0$ to show that that the generalized Nakayama conjecture holds for connected graded algebras. (Caution: Wilson writes "graded algebras" in that paper to mean what I have here been careful to call "connected graded algebras".)

The generalized Nakayama conjecture was originally stated by Auslander and Reiten, who observed that it implies the Nakayama conjecture. As I understand it, this in turn is equivalent to both of the Tachikawa conjectures being true. One would hope that Wilson's theorem then implies that the first and second Tachikawa conjectures are true for connected graded algebras, and in particular for monomial algebras, which is what you asked about.

I do not know if the truth of the generalized Nakayama conjecture *for a particular algebra* also implies the truth of the Tachikawa conjectures *for that same algebra*, when the truth of the generalized Nakayama conjecture in general is not known. Unfortunately I am not really an algebraist and I do not know the proofs of the implications between these conjectures well enough to attest to what happens when the generalized Nakayama conjecture is known only for a certain algebra or class of algebras! But I suspect others on MO who read this question know those proofs quite well, and perhaps they can complete this answer.

P.S. As I wrote, I am not really an algebraist, and I apologize if I have written anything foolish, which is quite possible.