Labor and Julia Gillard remain in polling purgatory, with neither able to shift voters’ antipathy toward them, according to a new round of polling from Essential Research.

Gillard’s personal approval rating remains near its all-time low at 32%, the same as a month ago. Her disapproval rating has fallen three points to 58%, leaving her with a 26-point net disapproval figure.

Tony Abbott, however, has picked up two points on approval to 38% and dropped two points on disapproval, to 50%, for his least-worst personal approval result since October. But the opposition leader still trails badly with female voters — only 30% of women approve of his performance and 52% disapprove.

And Abbott now only trails Gillard as preferred prime minister 36% to 38%; last month Gillard led 40-37%.

While the Coalition has lost two points on its primary vote to fall to 48%, Labor has failed to shift from its disastrous level of 31%. With the Greens remaining on 11%, the Coalition’s lead on a two-party preferred basis is 56-44%.

Meanwhile, Essential’s online panel of 1024 voters found support for the National Broadband Network has risen to its highest level ever, with 57% of voters backing the multi-billion dollar project and only 22% opposed to it. In February, support was 56-25%. For the first time ever, more Liberal voters support the NBN than oppose it, 42-40%.

Support is much stronger in areas where the NBN will be rolled out over the next three years (79%) than in areas that won’t get access until beyond then. Around 46% of voters said they would definitely or probably sign up for internet access via the NBN.

Essential also asked voters what responsibilities they feel should be controlled by different levels of government, or shared. Almost half (46%) of those polled thought improving the health system should be controlled by the Commonwealth, with only 42% saying it should be shared; 49% thought managing water should be a shared responsibility; and 45% of voters thought the Commonwealth should control the management of mining projects and mining profits, with only 30% saying it should be shared and 17% saying they should be solely controlled by the states.