Does such a weird convergent sequence exist?

Let $X$ be the Banach space of all $L^1$ functions $\mathbb{R}\rightarrow\mathbb{R}$. Take a convergent sequence $f_1,f_2,\ldots$ of functions from $X$. Let's call a sequence $g_1,g_2,\ldots$ of functions from $X$ a straightening of sequence $f_1,f_2,\ldots$ iff:

1. for each $n\geq1$ there exists $t_n$ such that $g_n(x)=f_n(x+t_n)$, and
2. for each $n>1$, this $t_n$ is chosen so that $g_n$ and $g_{n-1}$ are as close to each other as possible, i.e.$$\int_\mathbb{R}|g_{n-1}(x)-g_n(x)|dx=\min_{t\in\mathbb{R}}\int_\mathbb{R}|g_{n-1}(x)-f_n(x+t)|dx.$$

My question is: Is there a convergent sequence in $X$ such that all or at least some (these are the two variants of the question) of its straightenings are non-convergent?