Realization of the order-seven automorphism case with multiplicities (3,3)

Determine whether an order-seven automorphism of a fake projective plane can have a minimal resolution of its quotient whose two multiple fibers both have multiplicity $3$.

Background

For an order-seven automorphism of a fake projective plane, the minimal resolution of the quotient is a minimal elliptic surface with two multiple fibers. The possible multiplicity pairs listed in the paper are (2,3)(2,3), (2,4)(2,4), and (3,3)(3,3).

The first two possibilities are known to occur in the cited classification and computational results, whereas the paper singles out the (3,3)(3,3) case as unresolved. Its realization would clarify which of the symplectic action cases in Theorem 1.3 can occur algebraically.

References

However, whether the third case (i.e., both multiple fibers in $X$ have multiplicity $3$) is realized is unknown.

— Symplectic symmetries of a hypothetical exotic $CP^2$: a potential construction or a nonexistence proof  (2610.06746 - Chen, 5 Oct 2026) in Section 3.3, discussion of order-seven automorphisms of fake projective planes