Does "a pretty egg box" always mean "a pretty box of eggs" rather than "a box of pretty eggs"?

More precisely, is "adjective adjunct-noun head-noun" always interpreted as "adjective (adjunct-noun head-noun)" rather than "(adjective adjunct-noun) head-noun"?

If so, can "a pretty egg box" be punctuated so as to mean "a box of pretty eggs"?

Another more semantic example: "steel screwdriver box".

Does "a pretty egg box" always mean "a pretty box of eggs" rather than "a box of pretty eggs"?

More precisely, is "adjective adjunct-noun head-noun" always interpreted as "adjective (adjunct-noun head-noun)" rather than "(adjective adjunct-noun) head-noun"?

If so, can "a pretty egg box" be punctuated so as to mean "a box of pretty eggs"?

Does "a pretty egg box" always mean "a pretty box of eggs" rather than "a box of pretty eggs"?

More precisely, is "adjective adjunct-noun head-noun" always interpreted as "adjective (adjunct-noun head-noun)" rather than "(adjective adjunct-noun) head-noun"?

If so, can "a pretty egg box" be punctuated so as to mean "a box of pretty eggs"?

Does "a pretty egg box" always mean "a pretty box of eggs" rather than "a box of pretty eggs"?

More precisely, is "adjective adjunct-noun head-noun" always interpreted as "adjective (adjunct-noun head-noun)" rather than "(adjective adjunct-noun) head-noun"?

If so, can "a pretty egg box" be punctuated so as to mean "a box of pretty eggs"?

Another more semantic example: "steel screwdriver box".