Irreducible characteristic polynomial conjecture for matrix Kloosterman sums

Let $a$ be an $n\times n$ matrix over $\mathbb{F}_q$, with characteristic polynomial $P_a$ irreducible over $\mathbb{F}_q$, and let $\alpha\in\mathbb{F}_{q^n}$ be an eigenvalue of $a$. Denote by $K_n(a,\mathbb{F}_q)$ the matrix Kloosterman sum and by $K_1(\alpha,\mathbb{F}_{q^n})$ the scalar Kloosterman sum. Irreducible characteristic polynomial conjecture. If $p$ is large enough, then

$$K_n(a,\mathbb{F}_q)=(-1)^{n+1}q^{n(n-1)/2}K_1(\alpha,\mathbb{F}_{q^n}).$$

The proposed identity relates the matrix Kloosterman sum to a scalar Kloosterman sum over the degree-$n$ extension field in the regular semisimple case. The source does not provide evidence resolving this statement, and its status is therefore left open.