Is this a Boggle pad?

Question

Given a list of case-insensitive ASCII letter strings ("words"), output whether the entire list can be found on some four-by-four configuration ("board") of letter squares, in which no square can be used more than once in a word, and in which words are formed by moving from any square to any adjacent square including diagonally.

You do not need to worry that the combination of squares that would allow for the list actually appears in a Boggle game. The words need not be actual words in any language. You should assume that if Q appears on the board, it's on a square of its own (unlike in actual Boggle).

Standard loopholes are banned, standard I/O rules apply, and you're golfing.

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In the examples below, a slash signifies a newline.

Truthy examples

• auuuuooiiaaoiee, euiaaiueeuua, ooouuueee, eueueuoaoa — All are on the board auii/euoa/ueoa/euio
• swiop, byteba, ceropl, qoiz, aeis, lqoep — All are on the board abcb/rety/poiw/lqzs

Falsy examples

• swiop, byteba, ceropl, qoiz, aeis, lqoep, wybez — There are fifteen distinct letters (abceilopqrstwyz) and a single word has two bs, so that's the sixteenth square. Thus there's only one e. But the e has nine distinct letters (abcioprtz) adjacent to it, which is impossible.
• hskcbnzvgfa, lajdnzvfhgs, kajfbmxvshd, ldgckvbsanx — There are sixteen distinct letters (abcdfghjklmnsvxz). Of those, s is adjacent to all of abghkv, v is adjacent to all of bfgksxz, b is adjacent to all of cfmnsv, a is adjacent to all of fjklns, and f is adjacent to all of abghjv. But on an actual board only four squares are adjacent to more than five squares apiece.

Answer 1

Python 3, 215 bytes

def f(w):b=product(*[set("".join(w))]*16);return any(all(any(all(x-y in[1,-1,4,-4]for x,y in zip(q,q[1:]))for q in product(*[[x for x in range(16)if k[x]==c]for c in a]))for a in w)for k in b)
from itertools import*

Try it online!

This is never going to finish, so I can't really debug it or prove it works. I will try to find constant optimizations / test with reduced test cases, but if you spot a reason it would definitely fail, I will take it down / fix it.