Exact valuation conjecture for Somos-4 and Somos-5 sequences

Let $(\tau_n)$ be the terms of the Somos-$k$ sequence with $k\in\{4,5\}$, with parameters $\alpha=\beta=1$ and initial values

$$\tau_1=\cdots=\tau_k=1.$$

Set $d=2n-k-1$, and define

$$q=\begin{cases}\tau_n,&(k+1)\nmid n,\tau_n/2,&(k+1)\mid n.\end{cases}$$

Exact valuation conjecture. For the relevant integers $m$ and $l$, one has

$$q^{m+1}\mid\tau_l\quad\Longleftrightarrow\quad l=n+\left(\frac{q^m-1}{2}+kq^m\right)d.$$

The claim refines the known arithmetic divisibility pattern for the unit-initial-value Somos-4 and Somos-5 sequences. It is proposed conditionally on proving that, for odd primes, the multiples of a prime in these sequences are not divisible by exactly the same power of that prime; the supplied text does not establish the conjecture or specify the ranges of $m$ and $l$.