A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number of prime Euclid numbers?

A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number of prime Euclid numbers? The references on the papers in which people have tried partially answer the question are also accepted.

A number defined as the product of first $n$ prime numbers $+1$ is called Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number of prime Euclid numbers?

A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number of prime Euclid numbers?