Let $(X,d)$ be a complete metric space. Call a sequence $(x_n)\subseteq X$ a weakly Cauchy sequence in $X$ if there is some $y\in X$ such that $(d(y,x_n))_n$ is a Cauchy sequence in $\mathbb{R}$. It is clear from the estimate $$|d(y,x_m)-d(y,x_n)|\leqslant d(x_m,x_n)$$ that a Cauchy sequence in $X$ is a weakly Cauchy sequence. It is also true that a Cauchy sequence $(x_n)$ is bounded i.e. there is $z\in X$ such that $\sup_nd(z,x_n)<+\infty$.
Is it true that a bounded and weakly Cauchy sequence in $X$ is a Cauchy sequence in $X$?
If not, under what additional conditions is a weakly Cauchy sequence in $X$ a Cauchy sequence?