Uniqueness of tangent currents for positive closed currents

Let $X$ be a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a K\"ahler submanifold of dimension $l,$ and let $B\subset V$ be a piecewise $\mathcal{C}^2$-smooth domain. Let $T$ be a positive closed currents of bidegree $(p,p)$ in $X$ such that $T$ satisfies a mild reasonable assumption in a neighborhood of $\partial B$ in $X$ and that the $j$-th average mean $\nu_j(T,B,r)$ for every $j$ with $\max(0,l-p)\leq j\leq\min(l,k-p)$ converges sufficiently fast to the $j$-th generalized Lelong number $\nu_j(T,B)$ as $r$ tends to $0$ so that $r^{-1}(\nu_j(T, B,r)-\nu_j( T,B))$ is locally integrable near $r=0.$ Then we show that $T$ admits a unique tangent current along $B.$ A local version where we replace the condition of $T$ near $B$ by the conditions on a finite cover of $B$ by piecewise $\mathcal{C}^2$-smooth domains in $V$ is also given. When $T$ is a current of integration over a complex analytic set, we show that $\nu_j(T,B,r)-\nu_j(T,B)=O(r^\rho)$ for some $\rho>0,$ and hence this condition is satisfied. Our result may be viewed as a natural generalization of Blel-Demailly-Mouzali's criterion from the case $l=0$ to the case $l>0.$ The result has applications in the intersection theory of positive closed currents.