Minimal order of the recurrence for the parity-counting sequence

Determine whether the order-two recurrence operator L for the sequence a(n) counting standard Young tableaux of shape (n,n,n) with odd entry T(2,1) has minimal order, equivalently whether a(n) is a hypergeometric sequence.

Background

The paper derives an order-two linear recurrence with polynomial coefficients for a(n) and shows that the previously conjectured order-three recurrence is a left multiple of it. The authors report that exact nullspace computations exclude first-order operators up to coefficient degree 12, but explicitly note that this is evidence rather than a proof that no first-order recurrence exists.

A first-order polynomial-coefficient recurrence would establish that a(n) is hypergeometric. The paper leaves open whether the displayed order-two operator is genuinely minimal.