Winning-player conjecture for loopy cycles

Let $C(n,k)$ be a loopy cycle formed from a cycle $C_n$ by adding $k$ loops to consecutive vertices, with $n\geq 3$. Loopy-cycle winner conjecture. For $k=1$ and $k=2$, Player 1 wins. For $k\geq 3$, Player 2 wins when $k$ is odd and Player 1 wins when $k$ is even. The pattern is based on computational results, and the source does not provide a proof or resolution.