Spectral odd-cycle containment conjecture for the graph construction $R_k$

Let $k$ be a positive integer, let $G$ be a non-bipartite graph of order $n\geqslant 2k+1$, and let $R_k(H)$ denote the graph construction used in the source. Then

$$\lambda_1(G)\geqslant \lambda_1\left(R_k\left(K_{\lfloor(n-2k+1)/2\rfloor,\lceil(n-2k+1)/2\rceil}\right)\right).$$

Spectral odd-cycle containment conjecture. Under this condition, $G$ contains at least one cycle from

$$\{C_3,C_5,\ldots,C_{2k+1}\},$$

unless

$$G\cong R\left(K_{\lfloor(n-2k+1)/2\rfloor,\lceil(n-2k+1)/2\rceil}\right).$$

The conjecture extends the verified comparison in the source, which holds for $1\leqslant k\leqslant 5$, to all positive integers $k$. It is posed as an open problem in the supplied text.