This is not true. Synthetics move because they do not have sufficient elasticity to overcome the inter-string friction. Speaking of which, natural gut also does not stay grid straight as can be seen by many 90s pros picking at their stringbeds. Poly moving, however, tends to be a sign of death. That I agree with.
Those synthetic strings that move (ex. PSGD) usually stay straight for the first couple of sets. Then when they lose elasticity they start to move. This loss of elasticity is related to tension loss. When the string is less elastic, there is less tension, so there is definitely a relationship there. The playability goes down after that.