#Brachylog, 9 bytes

.v|:ladl1

Returns true or false or the empty list.

###Explanation

.v      Unify the output with the input if it's the empty list
|       Or
:la     Apply length to each element of the input
   d    Remove all duplicates
    l1  The length of the resulting list is 1

Brachylog, 9 bytes

.v|:ladl1

Returns true or false or the empty list.

Explanation

.v      Unify the output with the input if it's the empty list
|       Or
:la     Apply length to each element of the input
   d    Remove all duplicates
    l1  The length of the resulting list is 1

#Brachylog, 9 bytes

.v|:ladl1

###Explanation

.v      Unify the output with the input if it's the empty list
|       Or
:la     Apply length to each element of the input
   d    Remove all duplicates
    l1  The length of the resulting list is 1

#Brachylog, 9 bytes

.v|:ladl1

Returns true or false or the empty list.

###Explanation

.v      Unify the output with the input if it's the empty list
|       Or
:la     Apply length to each element of the input
   d    Remove all duplicates
    l1  The length of the resulting list is 1