The torus-knot $T_{3,m}$ superpolynomial prediction

Let $T_{3,m}$ be a torus knot, and let $\mathcal P(T_{3,m})$ denote its reduced superpolynomial. For $m=3k+1$ or $m=3k+2$, define the three polynomials $\mathcal P^{(0)},\mathcal P^{(1)},\mathcal P^{(2)}$ by the finite sums in the source statement. Torus-knot $T_{3,m}$ superpolynomial conjecture. The reduced superpolynomial has the form

$$\mathcal P(T_{3,m})=a^{2m-2}\mathcal P^{(0)}+a^{2m}\mathcal P^{(1)}+a^{2m+2}\mathcal P^{(2)}.$$

The explicit formulas give a prediction for the reduced triply graded theory of all torus knots of type $(3,m)$; the source gives no resolution status.