A.P. Mani and R.J. Stones' conjecture for the Tutte polynomial of complete graphs

Let $T(K_n;a,b)$ denote the Tutte polynomial of the complete graph $K_n$, and let $p$ be an odd prime and $k$ a positive integer. Let $u$ denote congruence modulo the indicated power of $p$. A.P. Mani and R.J. Stones' conjecture. If $n\geq p^k$, $a,b\in\mathbb{Z}$, and $b\not\equiv 1\pmod p$, then

$$T(K_n;a,b)\equiv\begin{cases} b^{\frac{\phi(p)}{2}-1} & \text{if }p\geq 3,\ n=p,\text{ and }a\equiv 1\pmod p,\\ b^{\frac{\phi(p^k)}{2}}T(K_{n-\phi(p^k)};a,b) & \text{otherwise} \end{cases}\pmod p.$$

If $n\geq p^k$, $a,b\in\mathbb{Z}$, and $b\equiv 1\pmod p$, then

$$T(K_n;a,b)\equiv\begin{cases} (n+a-1)^{p^k}T(K_{n-p^k};a,b) & n>p^k,\\ (a-1)^{p^k-1} & n=p^k \end{cases}\pmod {p^k}.$$

This conjecture refines the known congruence for $T_n(1,b)$ modulo prime powers by proposing recurrences for the complete-graph Tutte polynomial with general integer parameters $a$ and $b$. The supplied source does not establish a resolution, so the conjecture is recorded as open.