I am aware that the minimum (cardinality) vertex cover problem on cubic graphs ($3$-regular) graphs is $NP$-hard. Say positive integer $k>2$ is fixed. Has there been any computational complexity proof (aside from the $3$-regular result, note this would be $k=3$,) that shows the minimum (cardinality) vertex cover problem on $k$-regular graphs is $NP$-hard (e.g., $4$-regular)? Since $k$ is fixed, you aren't guaranteed the cubic graph instances needed to show the classic result I mentioned above.
Note that this problem would be straightforward to see is $NP$-hard from the result I mentioned at the start if we were to state that this were for any regular graph (since $3$-regular is a special case), we don't get that when $k$ is fixed.
Does anybody know of any papers that address the computational complexity of minimum (cardinality) vertex cover on a $k$-regular graph, when $k$ is fixed/constant? I have been having difficulties trying to find papers that address this (in the slew of documents that cover the classic result of minimum (cardinality) vertex cover on cubic graphs being $NP$-hard.)
Thank you so much!
EDIT (again): @R B gave a construction below!