Disjointness of parity-defined candidate values in the Fibonacci-average greedy sequence

Prove that the sets { n F_{2n+3} − (n − 1) F_{2n+2} : n ≥ 1 and n is even } and { F_{n+1}^2 + 2 : n ≥ 1 and n is odd } are disjoint, where F_m denotes the m-th Fibonacci number. This disjointness would ensure that the proposed closed-form expressions for the terms of the greedy sequence of distinct least positive integers whose first-n-term average is a Fibonacci number (for even and odd n, respectively) do not yield repeated values.

Background

The paper studies sequences arising from OEIS conjectures and, in Section 2.9, considers a greedily defined sequence where the average of the first n terms is a power of 2, proving a closed form. Motivated by a related OEIS conjecture (A248982), the authors then consider the analogous greedy sequence where the average of the first n terms is a Fibonacci number and propose closed-form expressions for the terms depending on the parity of n.

To validate that proposal fully, one must guarantee that the even-index and odd-index formulas never produce the same integer, since the sequence is required to consist of distinct least positive integers. The authors explicitly state they could not prove the required disjointness between the two parity-defined sets of values, leaving this as an unresolved question.