The multiplicity conjecture for Whitehead groups of $C_{p^n}\times C_{p^n}$

Let $p$ be an odd prime and let $i$ be a positive integer. For $n\geq 2i$, define

$$T_i(n)=(p-1)\left(2^{E(i)}p^{n-(\left\lfloor i/2\right\rfloor+2)}+(n-2i)p^{i-1}\right),$$

where $E(i)$ is $1$ if $i$ is even and $0$ if it is odd. Let $SK_1(\mathbb Z[C_{p^n}\times C_{p^n}])$ be the Whitehead group under consideration. Multiplicity conjecture. For an integer $n\geq 2$ and an odd prime $p$, the multiplicity of $C_{p^i}$, for $0<i<n$, in the decomposition of $SK_1(\mathbb Z[C_{p^n}\times C_{p^n}])$ as a product of cyclic groups is $T_i(n)$ if $2i\leq n$, and $T_{n-i}(2(n-i))$ otherwise. In particular, the multiplicity of $C_p$ is $T_1(n)=(p^{n-2}+n-2)(p-1)$, while the multiplicity of $C_{p^{n-1}}$ is always $T_1(2)=p-1$. The conjecture is motivated by explicit computations for $n=2,3,4,5,6$; the exponent is known to be $p^{n-1}$, so the multiplicity of $C_{p^n}$ is zero. The general decomposition remains conjectural.