Nonnegative Laurent phenomenon conjecture for Somos-5 sequences

Let $a_1,a_2,\dots$ be a Somos-5 sequence, meaning that every six consecutive terms satisfy

$$a_n a_{n+5}=a_{n+1}a_{n+4}+a_{n+2}a_{n+3}.$$

The Somos-5 Laurent positivity conjecture. Every term $a_n$ is a Laurent polynomial with nonnegative integer coefficients in the first five terms $a_1,\dots,a_5$.

The Laurent-polynomial property itself is known, but the conjecture asks for coefficientwise nonnegativity. It is presented as an analogue of the preceding conjecture and concerns the cancellation-free structure of the Somos-5 recurrence.