There are two layers underlying the issue: The first layer is, formally we can show every classical conservation law has a quantum counterpart through, say, canonical quantization(see Weinberg's QFT chap7). The second layer is, if we go beyond the formal level, it is really that we would **want **the conservation law to hold and strive to **make** it hold in the quantized case too, unless there are obstructions such as quantum anomalies.

On the surface of the issue, if we canonically quantize a classical Lagrangian, all conservation laws should be preserved, and this is fairly simple to see: the field operator equations have exactly the same forms of their classical versions, hence if any classical quantity has a vanishing 4-divergence, so will its quantized version.

However, reading between OP's lines, I suspect OP is having the misconception that a QFT can be defined by "a Lagrangian + a quantization recipe(canonical, path integral etc.)" This is not entirely true, a regularization and renormalization scheme must also be specified to make the QFT unambiguous. During this procedure, many symmetries are broken in the intermediate stages, and it is not a priori clear that the symmetries can all be restored after the procedure is finished.

The common practice is, through carefully controlling the regularize-renormalize procedure, we **define** a QFT preserving as many symmetries and conservations as possible. It is not always possible to preserve **all **the classical symmetries at hands, in such cases we will have the quantum anomalies.

Hence my earlier comment under your question:

My impression is that they will still be conserved separately in the quantized case, unless there are anomalies.