The threaded-digon complex conjecture for the flag Hilbert scheme

Let $a_1,\ldots,a_m,b\geq0$ and set $\bm{a}=(a_1,\ldots,a_m)$. A one-sided twisted complex of threaded digons is a chain of complexes obtained by increasing the number of crossings between the $b$-labeled strand and the threaded strands. Threaded-digon complex conjecture. There exists such a complex $\operatorname{TD}_b(\bm{a})$ satisfying: (i) $\operatorname{TD}_b(\bm{a})\simeq0$ if $b>a_1+\cdots+a_m$; and (ii) the partial trace

$$\operatorname{Tr}^{b}\bigl(\operatorname{TD}_b(1,\ldots,1)\otimes\mathbb{Q}[v_1,\ldots,v_b]\bigr)$$

categorifies the $b$th elementary symmetric function in the Jucys--Murphy braids on $m$ strands. Moreover, this complex should correspond to the $b$th exterior power of the tautological bundle on the flag Hilbert scheme $\mathrm{FHilb}_m(\mathbb{C}^2)$ under the Gorsky--Negut--Rasmussen conjecture. This is proposed as a categorified construction of the complexes predicted by the GNR conjecture.