The Macdonald superpolynomial norm conjecture

Let $\Lambda$ be a superpartition of fermionic degree $m$, and let $P_\Lambda(x,\theta;q,t)$ be the Macdonald superpolynomial. Let $\mathcal B\Lambda$ be the set of bosonic boxes, namely boxes whose row and column do not both end with a circle. Define

$$h^\downarrow_\Lambda(q,t)=\prod_{s\in\mathcal B\Lambda}\left(1-q^{a_{\Lambda^{\circledast}}(s)}t^{l_{\Lambda^*}(s)+1}\right),$$

and $h^\uparrow_\Lambda(q,t)=h^\downarrow_{\Lambda'}(t,q)$. Norm conjecture. The superspace scalar product satisfies

$$\left\langle\!\left\langle P_\Lambda(x,\theta),P_\Lambda(x,\theta)\right\rangle\!\right\rangle_{q,t}=(-1)^{\binom m2}q^{|\Lambda^a|}\frac{h^\uparrow_\Lambda(q,t)}{h^\downarrow_\Lambda(q,t)}.$$

The paper proves this conjecture in the stable sector $m\ge n$, but explicitly retains its conjectural status in the non-stable sector.