Number-theoretic conjecture for toric polynomial generators in the remaining even dimensions

Suppose $n$ is even and $n+1$ is not a prime power. Let $R_n(\varepsilon)$ denote the integer-valued expression used in the source for the relevant toric blow-up construction, with $\varepsilon\in\{2,\ldots,n-1\}$.

Number-theoretic conjecture. There exists an integer

$$\varepsilon\in\{2,\ldots,n-1\}$$

such that

$$\gcd(R_n(\varepsilon),n+1)=1.$$

This condition is proposed to guarantee that a sequence of blow-ups at torus-fixed points yields a smooth projective toric variety representing a polynomial generator of complex cobordism in the remaining even dimensions. The source describes the conjecture as unresolved and gives overwhelming numerical evidence for it.