Is this uniquely concatenable? In this challenge about prefix code, we learnt that prefix codes are uniquely concatenable.

That means, they can be joined together without separator, and without ambiguity.

For example, since [1,2,45] is a prefix code, I can join them together without separator as such: 1245245112145, and there will be no ambiguity.

However the converse is not always true.

For example, [3,34] is not a prefix code. However, I can do the same: 3333434334333, and there will be no ambiguity. You would just need a smarter parser.

However, [1,11] is not uniquely concatenable, because 1111 can be interpreted in 5 ways.

Goal

Your task is to write a program/function that takes a list of strings (ASCII) as input, and determine if they are uniquely concatenable.

Details

Duplicate counts as false.

Scoring

This is code-golf. Shortest solution in bytes wins.

Testcases

True:

[12,21,112]
[12,112,1112]
[3,4,35]
[2,3,352]


False:

[1,1]
[1,2,112]
[1,23,4,12,34]
[1,2,35,4,123,58,8]
05AB1E, 15 bytes
On the phone, so explanation will have to follow later. Code:
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Uses CP-1252 encoding. Try it online!.
Takes too much memory for the last test case, so that might not work...