Yes, this belongs to the theory which is called Additive Inverse Eigenvalue Problems. See, for example this paper, which seems to treat a very similar problem:

D.Paul Phillips, Some partial inverse eigenvalue problems: recovering diagonal entries of symmetric matrices, Linear Algebra and its Applications Volume 380, 15 March 2004, Pages 263-270,

and here is a survey:

Moody T. Chu, Inverse eigenvalue problems, SIAM Rev. Vol. 40, No. 1, pp. 1–39, March 1998

Your problem can be restated as follows: To a given symmetric matxix, can you add a diagonal matrix so that the result has eigenvalue $0$ with high multiplicity?

This belongs to the theory which is called Additive Inverse Eigenvalue Problems. See, for example this paper, which seems to treat a very similar problem:

D.Paul Phillips, Some partial inverse eigenvalue problems: recovering diagonal entries of symmetric matrices, Linear Algebra and its Applications Volume 380, 263-270,

however the exact statement you ask does not follow from this result, and I suppose that your problem is unsolved.

Here is a survey of such problems:

Moody T. Chu, Inverse eigenvalue problems, SIAM Rev. Vol. 40, No. 1, pp. 1–39.