Alternating signs of Φ_{2k+1}(0) for the trivial-seed polynomials

Determine whether, for the polynomial family Φ_n(z) defined by the formal solution f(z) = Σ_{n≥0} Φ_n(z) β^n of the delay Painlevé-I equation (T+1) f′ = (T−1) f^2 + 2β with shift operator Tφ(z) = φ(z+1), initial seed Φ_0(z) ≡ 0, and all integration parameters τ_i set to zero, the sequence of values Φ_{2k+1}(0) alternates in sign as k increases (i.e., sign(Φ_{2k+1}(0)) = (−1)^k).

Background

Section 4 introduces polynomials Φ_n(z) via a recursive construction arising from the formal series solution f(z) = Σ Φ_n(z) β^n to (T+1) f′ = (T−1) f^2 + 2β with Φ_0 ≡ 0. Setting all τ_i = 0 yields an explicit sequence Φ_n(z) with Bernoulli-type symmetry and various structural properties.

Immediately after listing several structural properties (including Φ{2k}(0) = 0 and certain symmetry relations), the authors note a sign-pattern phenomenon for the odd-index evaluations at zero, which they believe to hold but are currently unable to prove. This concerns the specific sequence {Φ{2k+1}(0)} and its sign behavior.