<p>As a first step towards a conjecture of Kahle and Newman, we prove that if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_142_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(T_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is a random 2-dimensional determinantal hypertree on <i>n</i> vertices, then <Equation ID="Equ13"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_142_Article_Equ13.gif" Format="GIF" Height="38" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \frac{\dim H_1(T_n,\mathbb {F}_2)}{n^2} \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfrac> <mrow> <mo>dim</mo> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </mfrac> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the 1-out 2-complex. Our proof relies on the large deviation principle for the Erdős–Rényi random graph by Chatterjee and Varadhan.</p>