The upgraded polynomiality conjecture for descendent invariants of K3 surfaces

Let $S$, $\beta$, $\delta_i$, and the normalized descendent invariant be as in the polynomiality conjecture, with $k_i,m_i\geq1$. For subsets $I_x\subset\\{1,\ldots,x\\}$, where $x$ ranges over $r,s,t,u$, fix the indices whose labels are not in the corresponding subsets. Upgraded polynomiality conjecture. There exists a polynomial $p$ of degree $\beta^2+2-2u-t+r$ such that, when the varying indices $k_i,\ell_i,m_i,n_i$ with $i$ in $I_r,I_s,I_t,I_u$ satisfy the same polynomial range as above, the normalized invariant equals

$$p\left((k_i)_{i\in I_r},(\ell_i)_{i\in I_s},(m_i)_{i\in I_t},(n_i)_{i\in I_u}\right).$$

This strengthens the preceding conjecture by asserting polynomiality in any selected subset of descendent indices while the remaining indices are held fixed.