  Let $p$ be an odd prime and $d = p^{\tau}(p-1)$. In the spirit of Aritn's
conjecture, consider the system of two diagonal forms of degree $d$ in $s$
variables given by \begin{equation*}\begin{split} a_1x_1^d + \cdots + a_sx_s^d
= 0\\ b_1x_1^d + \cdots + b_sx_s^d = 0 \end{split} \end{equation*} with $a_i,
b_i \in \mathbb{Q}_p$. For $s > 2 \frac{p}{p-1}d^2 - 2d$, this paper shows that
this system has a non-trivial $p$-adic solution for every $\tau \ge 3, p \ge
7$, and for every $\tau = 2, p \ge \frac{C}{2}+4$, where $C \le 9996$.
Moreover, for $s > (2\frac{p}{p-1} + \frac{C-3}{2p-2})d^2 - 2d$, this system
will have a non-trivial $p$-adic solution for every $\tau = 1, p \ge 5$.
