The short answer is yes.
We assume that the original Schrödinger PDE was introduced for an isolated and bounded quantum system (microscopic or macroscopic) in infinite free space (-∞
In classical mechanics, where the solution is stimulated or forced via boundary conditions and/or a source term (BC and S), the solution can be symmetric or not depending on the presence of symmetry in BC and/or S.
On the other hand, in quantum mechanical problems where there is neither BC nor S, the solution is completely spontaneous and the symmetry of the solution is essential.
It is worth mentioning that in quantum mechanics there is a connection between symmetries and conservation laws.
Many physicists and mathematicians believe that quantum mechanics explains why Newton's laws of motion are good enough for classical and quantum mechanics.
Additionally, Newton's third law states that for every action, there is an equal and opposite reaction which can be translated into space-time symmetry (xt).
Furthermore, the B-matrix strings predict a symmetric solution for almost all quantum mechanical situations.
Figure 1 shows, via matrix chains B, the spatial distribution of the wave function ψ in a 1D infinite potential well. (with maximum probability)