Minimum spanning tree of a connected induced subgraph I'm doing an online course in which I'm struggling with the following multiple-choice question:

Suppose $ T $ is a minimum spanning tree of the connected graph $ G $.
Let $ H $ be a connected induced subgraph of $ G $. (I.e., $ H $ is
obtained from $ G $ by taking some subset $ S \subseteq V $ of
vertices, and taking all edges of $ E $ that have both endpoints in $
&gt; S $. Also, assume $ H $ is connected.) Which of the following is true
about the edges of $ T $ that lie in $ H $? You can assume that edge
costs are distinct, if you wish. [Choose the strongest true
statement.]

For every $ G $ and $ H $, these edges form a minimum spanning tree of $ H $
For every $ G $ and $ H $, these edges are contained in some minimum spanning tree of $ H $
For every $ G $ and $ H $ and spanning tree $ T_H $ of $ H $, at least one of these edges is missing from $ T_H $
For every $ G $ and $ H $, these edges form a spanning tree (but not necessary minimum-cost) of $ H $


I don't understand why option 4 is not correct; the hint given is as follows:

Suppose G is a triangle and H is an edge.

Suppose that G is a triangle with nodes 1, 2, and 3, all connected, and we choose the subgraph H from nodes 1 and 2, thus including only the edge (1,2). That edge then forms a minimum spanning tree of those two nodes, no?
Incidentally, the answer

For every G and H, these edges for a minimum spanning tree of H

is also incorrect.
Suppose that $G$ is the triangle on $\{1,2,3\}$ (with arbitrary edge weights), that $T$ is $\{\{1,2\},\{1,3\}\}$ (without loss of generality), and consider $H = \{\{2,3\}\}$, which is induced by $S = \{2,3\}$. No edges of $T$ lie in $H$, and in particular these edges do not constitute a spanning tree of $H$.