  The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and
Lison\v{e}k states that by inserting all cyclic permutations of some initial
blocks of 2's into the multiple zeta value $ \zeta(1,3,\ldots,1,3) $ and
summing, one obtains an explicit rational multiple of a power of $ \pi $.
Hoffman gives a conjectural identity of a similar flavour concerning $ 2
\zeta(3,3,\{2\}^m) - \zeta(3,\{2\}^m,(1,2)) $.
  In this paper we introduce the 'generalised cyclic insertion conjecture',
which we describe using a new combinatorial structure on iterated integrals --
the so-called alternating block decomposition. We see that both the original
BBBL cyclic insertion conjecture, and Hoffman's conjectural identity, are
special cases of this 'generalised' cyclic insertion conjecture. By using
Brown's motivic MZV framework, we establish that some symmetrised version of
the generalised cyclic insertion conjecture always holds, up to a rational;
this provides some evidence for the generalised conjecture.
