I would not try to "work backwards" from your estimates of heritability to infer anything about the phenotypic values of the parents. That isn't particularly useful. The values that you have estimated are the proportions of the observed phenotypic variances, for this trait among the progeny tested, that are heritable and most useful for breeding (being additive). To me, those values are only useful for making informed decisions about when and how to perform selection for the trait.

Thanks.

From my mixed model I also obtain the reliability of the breeding values (ie the squared correlation between estimated and true breeding values, ie the "CD" of animal breeders, computed from prediction error variance and inbreeding coefficients). I read in a powerpoint of Bruce Walsh that "the extent to which the

reliability exceeds h2 is a measure of how much information is added by relatives". So I wondered if I could use my results to say that mass selection among the parents would not be efficient (because h²A and h²B are low) while selection based on progeny tests is very efficient (because CD are high). I don't know if it's correct to make such a comparison.

Generally speaking, if your heritability is low then selection for the trait on the basis of single plants is unlikely to be very effective. Progeny testing will improve your ability to effectively select for the trait, because it is essentially a form of replication. Replication allows estimation and partitioning of the environmental component of phenotypic variance. With the environmental component partitioned, you will have much better estimates of your breeding values and be able to make more effective selection. This is assuming that the reason for your low NS heritability is environmental, rather than non-additive genetic. If it's non-additive genetic, I don't think progeny testing will improve your selection much, though I haven't thought that all the way through.