Romanovski polynomials, Gegenbauer connections, and $\mathrm{su}(1,1)$ ladder structures

We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters $β_K=-K$ and $α_K=c/(K+1)$. We write out the parameter preserving first order ladder relations at fixed $(α,β)$, with real constants. Combining them returns the three-term recurrence; iterating them builds $R_