mod $p$ non-vanishing conjecture for the WRT function

Let $M$ be an integral homology sphere and let $p$ be a prime. Its mod $p$ WRT function is

$$\tau _p(M)=\widetilde{\tau }_p(M)\colon\mathbb{F}_p^\times\to\mathbb{F}_p.$$

mod $p$ non-vanishing conjecture. The function $\widetilde{\tau }_p(M)$ is nowhere zero on $\mathbb{F}_p^\times$. The source says this is supported by computer calculations and gives no proof.