<p>Let <i>T</i>_<i>n</i> be a 2-dimensional determinantal hypertree on <i>n</i> vertices. Kahle and Newman conjectured that the <i>p</i>-torsion of <i>H</i>₁(<i>T</i>_<i>n</i>, ℤ) asymptotically follows the Cohen–Lenstra distribution. For <i>p</i> = 2, we disprove this conjecture by showing that given a positive integer <i>h</i>, for all large enough <i>n</i>, we have <Equation ID="Equ1"> <EquationSource Format="TEX">\({\mathbb P}(\dim H_{1}(T_{n},{\mathbb F}_{2}) \geq h) \geq {{e}^{-200h} \over (100h)^{5h}}.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>dim</mi> <mspace width="thinmathspace" /> <msub> <mi>H</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow> <mfrac> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mn>200</mn> <mi>h</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>100</mn> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mn>5</mn> <mi>h</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </math></EquationSource> </Equation></p><p>We also show that <i>T</i>_<i>n</i> is a bad cosystolic expander with positive probability.</p>