Self-injectivity of quotients of polynomial rings by zero-dimensional ideals

Question

Let $A = \Bbbk[x_1, \dots , x_n]$ for some field $\Bbbk$, and let $I$ be an ideal contained $(x_1, \dots, x_n)$.

If $A/I$ is finite dimensional, is it necessarily self-injective?

This answer seems to indicate that if we pass to the completion then the answer is "yes". Can we then infer (using associated graded etc.) that the same must be true of $A/I$?