In the original sentence, distributed can be interpreted either as part of a passive voice construction or as a predicate past participle adjective.
To understand it as passive, supply a distributor — an agent:
The board thought that the shares were unjustly distributed among the heirs [by the executors].
To understand it as a predicate adjective, look at a present tense version:
The board thinks that the shares are unjustly distributed among the heirs.
The test interprets it as part of a passive voice construction. The sentence then follows the rules for making an “impersonal passive,” or the passive voice with reporting verbs, as seen here:
- subject + passive reporting verb + to-infinitive
In this pattern we start with the subject of the reported clause, which is
followed by the passive reporting verb and the to-infinitive form of
the verb in the reported clause:
knows that my grandfather likes red wine.
grandfather is known to like red wine.
The reporting verb (is known) is in the same tense as it was in the
active sentence (knows). The type of to-infinitive we use (to
like) corresponds to the temporal relationship between the action of
reporting and the reported event. This temporal relationship can be of
two basic types:
The reporting and the reported event happen simultaneously (as in the
The reported event happens before the reporting.
Grammaring — Passive voice with reporting
Here’s as example of a reported event happening before the reporting:
Everybody thought that my grandfather lived in France when he was a child.
My grandfather was thought to have lived in France when he was a child.
This is like your sentence; your reported event (distribution) happened before the reporting:
The board thought that the shares were unjustly distributed among the heirs.
The shares were thought to have been unjustly distributed among the heirs.
(And now you have a full-on double passive.)