Existence of exceptional q=3 hemisystems and distance-regular graphs

Determine whether the Hermitian polar space (2d−1,3²) admits hemisystems with respect to (d−2)-spaces for d>2, and whether distance-regular graphs with classical parameters (d,−3,−2,−((−3)^d+1)/2) exist for d>2.

Background

The non-existence result proved in the paper excludes hemisystems with respect to (d−2)-spaces only when q≠3. The case q=3 is explicitly left unresolved. The authors further connect this geometric question to the existence of distance-regular graphs with the specified classical parameters. They note that for d=2 there is a unique graph with these parameters, the (Sims-)Gewirtz graph, but do not resolve existence for d>2.

References

This paper did not deal with the case q=3. Does (2d-1,32) admit hemisystems with respect to the (d-2)-spaces? More generally, do distance-regular graphs with classical parameters \Big(d,-3,-2,-\frac{(-3)d+1}2 \Big) exist for d > 2?

— A note on the classification of classical distance-regular graphs of negative type and the non-existence of hemisystems  (2511.15280 - Adriaensen et al., 19 Nov 2025) in Section “Open Problems”, item 2