unknown op-amp current leak
U1 (LTC3105)   

. FB    = 0.53V      AUX  =  1.49V   
  LDO   = 1.49V      V_OUT = 1.49V  
  FBLDO =    0V      PGOOD = open
  SHDN* = open       SW    = 1.07V   
  MPPC  = 1.39V      V_IN  = 1.07V 


If PSU off ,  V_IN on U3 is open  and C4=3F/2.7V
  - V_OUT/(R4+R5//R6)=1.49/(300k+560k//620k)=2.48uA
  - V_OUT/(R1+R2)=1.49/(1M+1.65M) = 0.56uA
  - above = 3uA total
  - supercap discharges with 12uA.
  - D9 leakage = ? with Vf= 1.49-1.44=0.05V    


REMOVE D9 to isolate  then remove any other paths for causes of leakage


U2 (MAX9015) 

. REF   = 1.23V      n.c.
  IN-   = 1.23V      Vcc   = 1.44V  
  IN+   = 0.72V      OUT   = 0V 
  Vee   =    0V      n.c.


U3 (LTC3525) 

  SHDN    = 0V       SW    = 1.49V
  GND     = 0V       GND   = 0V  
  V_IN    = 1.49V    V_OUT = 0V


PSU (to simulate solar cell):   

 1.10V 30mA Supercap: 
 1.49V discharging at 44uA