Quadratic average-degree conjecture for contractible edges

An edge $e$ of a $k$-connected graph is $k$-contractible if contracting $e$ leaves a $k$-connected graph. The average degree is the average of the vertex degrees.

Quadratic average-degree conjecture. There exists a constant $c$ such that every finite $k$-connected graph of average degree at least $c k^2$ admits a $k$-contractible edge.

Known bounds give an upper bound of order $k^2\log k$ for the relevant average degree, while examples show a quadratic lower-order scale, so the conjecture asks whether the logarithmic factor can be removed. The source gives no resolution evidence.