  The multi-bubble isoperimetric conjectures in $n$-dimensional Euclidean and
spherical spaces from the 1990's assert that standard bubbles uniquely minimize
total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q
\leq n+2$. The double-bubble conjecture on $\mathbb{R}^3$ was confirmed by
Hutchings-Morgan-Ritor\'e-Ros (and later extended to $\mathbb{R}^n$). The
double-bubble conjecture on $\mathbb{S}^n$ ($n \geq 2$) and the triple- and
quadruple- bubble conjectures on $\mathbb{R}^n$ and $\mathbb{S}^n$ (for $n \geq
3$ and $n \geq 4$, respectively) were recently confirmed in our previous work,
but the approach employed there does not seem to allow extending these results
further.
  In this work, we confirm the quintuple-bubble conjecture on $\mathbb{S}^n$
($n \geq 5$), and as a consequence, by approximation, also the quintuple-bubble
conjecture on $\mathbb{R}^n$ ($n \geq 5$) but without the uniqueness assertion.
Moreover, we resolve the conjectures on $\mathbb{S}^n$ and on $\mathbb{R}^n$
(without uniqueness) for all $q \leq n+1$, \emph{conditioned} on the assumption
that the singularities which appear at the meeting locus of several bubbles
obey a higher-dimensional analogue of Plateau's laws. Another scenario we can
deal with is when the bubbles are full-dimensional (``in general position"), or
arrange in some good lower-dimensional configurations.
  To this end, we develop the spectral theory of the corresponding Jacobi
operator (finding analogies with the quantum-graph formalism), and a new method
for deforming the bubbles into a favorable configuration. As a by-product of
our analysis, we show that the Jacobi operator on a minimizing configuration
always has index precisely $q-1$. Several compelling conjectures of PDE flavor
are proposed, which would allow extending our results to all $q \leq n+1$
unconditionally.
