  Let $g$ be a Riemannian metric for $\mathbf{R}^d$ ($d\geq 3$) which differs
from the Euclidean metric only in a smooth and strictly convex bounded domain
$M$. The lens rigidity problem is concerned with recovering the metric $g$
inside $M$ from the corresponding lens relation on the boundary $\partial M$.
In this paper, the stability of the lens rigidity problem is investigated for
metrics which are a priori close to a given non-trapping metric satisfying
"strong fold-regular" condition. A metric $g$ is called strong fold-regular if
for each point $x\in M$, there exists a set of geodesics passing through $x$
whose conormal bundle covers $T^*_{x}M$. Moreover, these geodesics contain
either no conjugate points or only fold conjugate points with a non-degeneracy
condition. Examples of strong fold-regular metrics are constructed and are
expected to be generic. Our main result gives the first stability result for
the lens rigidity problem in the case of anisotropic metrics which allow
conjugate points. The approach is based on the study of the linearized inverse
problem of recovering a metric from its induced geodesic flow, which is a
weighted geodesic X-ray transform problem for symmetric 2-tensor fields. A key
ingredient is to show that the kernel of the X-ray transform on symmetric
solenoidal 2-tensor fields is of finite dimension. It remains open whether the
kernel space is trivial or not.
