Increasing muscular endurance only makes you somewhat stronger. Increasing strength on the other hand also increases endurance. So no the correlation does not go both ways.
The reason for this is:
Henneman's size principle
Muscle fibers are grouped together in motor units. There are typically 3 to 15 muscle fibers in each motor unit.
All motor units in a muscle receives the same electrical signal from the brain. The smaller motor unit only requires a weak electrical signal to engage. They contain only type I fibers: slow enduring. The larger motor units requires a stronger electrical signal to engage. They contain mostly only type II fibers: fast and strong but not enduring. When a motor unit engage all muscle fibers in that motor unit contract. The brain starts by sending a weak electrical signal first. This recruits only the smallest motor units and therefore only a small force is generated. It then increases the strength of the electrical signal gradually (but very fast) and more and more motor units are recruited and more and more force is generated. It does this until the produced force matches the requirements. This wonderful mechanism enables the same muscles that can be used to lift heavy weights to eg. perform brain surgery. As mentioned the motor units are recruited from the smallest to the largest based on the force demands placed on the muscle. This is Henneman's size principle. It has been verified in experiments using EMG measurements of muscle activity. It is therefore a scientific fact and not a postulate.
The size principle has one very important practical consequence: in order to recruit all motor units and train the whole muscle a high force must be exerted. A low resistance (such as 20 RM) do not recruit all motor units and therefore do not train the whole muscle.
Therefore the answer to this question is no, the correlation does not go both ways. Training with high force (heavy weights or medium weights fast) trains the whole muscle. That includes the enduring type I muscle fibers and the stronger but not enduring type II muscle fibers. Training with high force therefore makes you both strong, fast and enduring. Training with light weights on the other hand only train the enduring type I muscle. It therefore makes you enduring and somewhat strong.
This does not necessarily imply that one should train with heavy weights all the time. However it implies that one should train with heavy weights (<=5 repetitions) some of the time.
Also note that even though the size principle shows that the causality must be stronger from strength to endurance than the other way, this does not mean that the causality does not break down at sufficiently large reps. And it does. I am pretty sure that doing backsquat sets of 5 or less alone does not enable you to a 100 bodyweight squats in row. No matter how large your 1RM backsquat becomes. If you need to do 100 bodyweight squats in a row you should probably also train with many repetitions and high volume ala the hundredpushups program. Here we can refer to the SAID principle, but in order to really understand why we would have to take an in depth look at the specific adaptions caused by endurance training. Probably endurance training cause more efficient generation and use of energy and oxygen in muscle cells.