Is it decidable whether the output length of a transducer is bounded by the input length? The transducers considered here are those Wikipedia calls finite state transducers.  The behavior of a transducer $T$, that is, the relation it computes, is written $[T]$: a word $y$ is an output for $x$ iff $x[T]y$.

Question: Is the following problem decidable:


  Given: A transducer $T$ and a regular language $L$
  Decide: Does it hold that $\forall x \in L$, $\forall y$ a word, $x[T]y$ implies that $|y| \leq |x|$?


I'm looking for nontrivial analysis/solvable subcases, reduction to known problems and/or related refs. (right now not even sure it is decidable in general...?)

Motivation: this problem was inspired by analysis/inquiry into automated theorem proving of number theoretic problems in general and a highly-studied one, Collatz conjecture, in particular.
The other contributor deleted his answer, maybe to let me extend my above comment, so here it is.

Let $T$ be a possibly nondeterministic transducer, and $L$ be a regular language.  Modify $T$ into a transducer $T'$ that checks that its input is in $L$ (by, e.g., changing the state set into the Cartesian product of the state sets of  $T$ and $L$, and modifying the transition function so that the $L$ part of the states is properly updated, while retaining the behavior of $T$.)

A branch of $T'$ is a sequence $\rho_1 C_1\rho_2 C_2 \cdots \rho_n C_n \rho_{n+1}$ such that $\rho_1\rho_2 \cdots \rho_{n+1}$ is an accepting simple path in $T'$, and each $C_i$ is a strongly connected component of $T'$ the states of which include the destination of $\rho_i$ (and the origin of $\rho_{i+1}$).  The branch is tame if:


The input length of the path $\rho_1\rho_2\cdots\rho_{n+1}$ is greater than or equal to its output length;
For any $i$, any simple cycle in $C_i$, the input length of the cycle is greater than or equal to its output length.


Fact: $\big[$ For any $x, y$, $x[T']y$ implies $|y| \leq |x|$ $\big]$ iff all branches are tame.

The proof is rather immediate.  The latter property being decidable (as the number of branches is bounded, and the number of simple cycles too), this shows that the problem of the question is decidable.