The $\widetilde{E}_7$ degree-$\pm(2,1)$ uniqueness conjecture

Let

$$\mathcal{A}\left(\left(\binom{-1}{-1},\binom{0}{0},\binom{-1}{-1},\binom{0}{0},\binom{1}{0},\binom{0}{0},\binom{1}{0},\binom{0}{1}\right),\widetilde{E}_7\right)$$

be the graded cluster algebra considered for affine type $\widetilde{E}_7$, with two-dimensional degree vectors. $\widetilde{E}_7$ degree-$\pm(2,1)$ conjecture. The degrees $(2,1)$ and $(-2,-1)$ correspond to only one variable each. The paper proves that several other degrees contain infinitely many variables; the asserted uniqueness in these two degrees is supported by computation but remains open.