No… I found 44442 solutions.
The highest one is:
11696519: {1299541, 1299653, 1299673, 1299553, 1299583, 1299637, 1299647, 1299601, 1299631};
If you are interested in the other solutions(with only consecutive primes(generating a sequence containing also all non-consecutive-prime solutions would take ages with brute force)) you can find a text document with all 44442 of them here
I don't think there would an end in the sequence. With infinitely many primes there should be a new number in the sequence when you search through enough primes.
However the sequence with consecutive primes only might end at some point, but it would certainly be after much more then 44442.
Update: I let the program run even further. I now got 351166 solutions.
The highest one is:
139372413: {15485807, 15485801, 15485857, 15485837, 15485773, 15485843, 15485849, 15485783, 15485863};This starts to look like prime k-tuplets; in our case 9-tuplets.
I started reading on the concept of admissible 9-tuplets, and why some patterns are not admissible.
This is the fun of pushing what at first appears to be a silly puzzle ;-)
I made the problem up myself as I was teaching a small group how to solve the simpler problem of placing the numbers 1 to 9 in the same triangle. That puzzle has a neat insight that cuts the solutions down enormously.