Are SNIa a two-component family?

In astro-ph/0803.1130, Dahlen, Strolger & Riess report on their HST survey for high redshift SN Ia. Their results strengthen the argument that the SNIa rate drops at high redshift. The immediate interpretation of this is that SN Ia only start after some long delay. Their favourite model has a delay time of 3.7 Gyr and is shown as the solid line in the Figure. The solid points in the Figure shows the observed SN Ia rate.

Now the problem is that a number of groups have recently argued, fairly convincingly, that SN Ia at lower redshift clearly fall into two categories: A group of delayed explosions and a group of "prompt" explosions - ala SN II. Three such models are shown as non-solid lines in the Figure and the match to the observations fails to excite :) What gives? The authors argue that dust obscured star formation might explain the differences, but it is not settled.