Conjectured exact value of the Ramsey number of a book graph versus an odd cycle

Let $B_n$ denote the book graph with $n$ pages and let $C_m$ denote the cycle of odd length $m$. The Ramsey number $R(B_n,C_m)$ is the least integer such that every red-blue edge-colouring of a complete graph of that order contains a red copy of $B_n$ or a blue copy of $C_m$.

The authors' conjecture. For each integer $n\geq 3493$, if $n\geq 2m+499$ and $m\geq 7$ is an odd integer, then

$$R(B_n,C_m)=2n+3.$$

The paper presents this as a conjecture motivated by the lower bound and notes that an earlier theorem partially confirms it for larger values of $n$. The authors also state that they proved the analogous claim for the subgraph $\mathbb{K}_{2,n}$ of $B_n$; the exact Ramsey-number assertion remains open in the supplied text.