Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, can be endowed with a riemannian metric of negative sectional curvature. Pansu mentions in a survey (4/TSG_1985-1986_*4*_101_0/TSG_1985-1986_*4*_101_0.pdf">http://archive.numdam.org/ARCHIVE/TSG/TSG_1985-1986_*4*/TSG_1985-1986_*4*_101_0/TSG_1985-1986_*4*_101_0.pdf) that he doesn't know how to make Gromov and Thurston's calculation in the general case. Is it possible to generalize ?

Namely let $M$ be a compact manifold endowed with a riemannian metric of negative sectional curvature. Suppose $X \longrightarrow M$ is a ramified covering of finite degree, along a compact submanifold of codimension $2$. Can $X$ be endowed with a riemannian metric of negative sectional curvature ? Can a counterexample be found when the locus of ramification is not totally geodesic ? Can one hope a pinching of the curvature like Gromov and Thurston ? Is the covering, in any reasonable sense, "more negatively" than the base ?