Hi, 

The following formulas are examples of non-positive structure coefficients
for non-symmetric cases which are easily verified by the algorithm presented
in Leclerc's paper "Dual Canonical Bases, Quantum Shuffles, and q-characters" 
or quagroup package in GAP4.

Professor Masaki Kashiwara told me that he has known such non-positive 
structure coefficient for $G_2$ since Shigenori Yamane found it in 1994
as treated in his master thesis at Osaka University (written in Japanese).
You can see similar negative coefficients in at least case $A_{2n}^{(2)}, D_{n+1}^{(2)}$.
Anyway, conjecture 52 in Leclerc's paper is false 
(I already told Professor Leclerc about it).

Shunsuke Tsuchioka

Notation: $G(i_1,\cdots,i_n)$ stands for the canonical basis element
corresponds to a crystal element 
$b(i_1,\cdots,i_n)=\tilde{f}_{i_n}b(i_1,\cdots,i_{n-1})=\cdots$. 

$G_2$ (1 is the short root) : 
$f_2 G(121112211)
= G(1211122211)
  + [2]G(1111222211)
  + G(2111112221)
  + [2]G(1211112221)
  + G(1111122221)
  - G(1112211122)
  + [2]G(1122111122)$

$C_3$ (1,2 are short roots) :
$f_3 G(23122312)
= [2]G(222333121)
  + [2]G(312222331)
  + [2]G(231222331)
  + [2]G(122223331)
  + G(231223312)
  + [2]G(122233312)
  - G(223112233)
  + [2]G(231122233)$


$B_4$ (1,2,3 are long roots) :
$f_1G(4342341234)
= [2]G(43344423211)
  + [2]G(43423443211)
  - G(44233443211)
  + [2]G(43423344211)
  + [2]G(43423442311)
  + [2]G(34234442311)
  + [2]G(43422334411)
  + G(43423412341)$